Proof
We prove the statement by induction on
. For
,
the statement is true. Suppose now that the statement is true for less than
vectors. We consider a representation of
, say
-
![{\displaystyle {}a_{1}v_{1}+\cdots +a_{n}v_{n}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/679c2c9479da275f59fbffe2a776e41e143bf455)
We apply
to this and get, on one hand,
-
![{\displaystyle {}a_{1}\varphi (v_{1})+\cdots +a_{n}\varphi (v_{n})=\lambda _{1}a_{1}v_{1}+\cdots +\lambda _{n}a_{n}v_{n}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5220d931ad8f81ee18b7193574988bf375e7d49)
On the other hand, we multiply the equation with
and get
-
![{\displaystyle {}\lambda _{n}a_{1}v_{1}+\cdots +\lambda _{n}a_{n}v_{n}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f75c2b73dda79278e4c1ac04d5b5aa4252d96e0)
We look at the difference of the two equations, and get
-
![{\displaystyle {}(\lambda _{n}-\lambda _{1})a_{1}v_{1}+\cdots +(\lambda _{n}-\lambda _{n-1})a_{n-1}v_{n-1}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60af08dacb7b9e5218681ba634edb1a658dfdbaf)
By the induction hypothesis, we get for the coefficients
,
.
Because of
,
we get
for
,
and because of
,
we also get
.